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Let \(h:\mathbb {R}\mapsto \mathbb {R}_+\)  be a strictly convex function and \(\mu ,\nu \in {P}(\mathbb {R})\)  be probability measures. Consider the cost \(c(x,y)=h(y-x)\)  and suppose that the optimal transport cost is finite. Then, the optimal transport problem has a unique solution given by \(\gamma _{mon}\) , the monotone coupling between \(\mu \) , and \(\nu \) .
